- Email: [email protected]
Electric dipole moments and b-τ unification in the presence of an intermediate scale in SUSY grand unification
s.__
21 November 1996
E!ld 33
ELSEVIER
PHYSICS
LETTERS B
Physics Letters B 388 (1996) 605-610
Electric dipole moments and b--7 unification in the presence of an intermediate scale in SUSY grand unification N.G. Deshpande a, B. Duttaa, E. Keith b a Institute of Theoretical Science, University of Oregon, Eugene, OR 97403, USA b Department of Physics, University of California, Riverside, CA 92521, USA
Received 8 June 1996; revised manuscript received 25 July 1996 Editor: M. Dine
Abstract We show that an intermediate gauge symmetry breaking scale can be a significant source of electric dipole moments for the electron and neutron in supersymmetric grand unified theories. New phases, similar to that of the CKM matrix, appear which do not arise from the supersymmetry (SUSY) breaking operators. To illustrate, we choose some grand unified SUSY models having an intermediate gauge symmetry breaking scale with some attractive features. We also show how well the b--7 unification hypothesis works in this class of models. PACS: 11.3O.Pb; 12.1O.Kt; 12.6O.J~; 14.60.Cd Keywords: Supersymmetry; Grand unification; Intermediate scale; Dipole moments
Supersymmetric grand unified theories (GUTS) with intermediate gauge symmetry breaking scales are attractive because they resolve a few long-standing problems and possess some desirable phenomenological features. For example, in models where the intermediate breaking scale MI N 10’“-10’2 GeV, one can naturally get a neutrino mass in the interesting range of N 3- 10 eV, which could serve as hot dark matter to explain the observed large scale structure formation of the universe [I]. The window N 10’“-10’2 GeV is also of the right size for a hypothetical PQ-symmetry to be broken so as to solve the strong CP problem without creating phenomenological or cosmological problems [ 21. Models which allow even lower intermediate gauge symmetry breaking scale, e.g. MI N 1 TeV are also interesting since they predict relatively light new gauge fields, as for example Sum charged gauge bosons WR. In all these intermediate 0370-2693/96/$12.00
scale models, lepton flavor violation is predicted [ 31 which may be close to the current experimental limit, and hence could provide a signal of such models. In this letter, we point out another special feature of these intermediate scale models: they can give rise to detectable amounts of electric dipole moments (EDMs) to the electron and neutron. This feature does not depend on the nature of the ultimate unifying group. We will always assume that supersymmetry is broken via soft breaking terms introduced at a super high scale. We shall assume that the soft breaking terms at the high scale at which they are introduced are flavor blind and CP invariant. It has already been shown [4-61 for SUSY SO( 10) models without an intermediate scale that there could be significant amounts of EDM for the electron and neutron. However for this to arise, the universal boundary condition for the soft SUSY breaking terms has to be imple-
Copyright 0 1996 Elsevier Science B.V. All rights reserved.
PII SO370-2693(96)01176-S
606
N.G. Deshpande et al./Physics
mented at a scale higher than the GUT scale MG such as the reduced Planck or string scales. Consequently, the EDMs for the electron and neutron are not expected to be produced in such a manner in SUSY GUTS with the attractive feature of gauge unification taking place at the string scale. The models with an intermediate gauge symmetry breaking scale however give rise to electron and neutron EDMs, irrespective of whether a universal soft SUSY breaking boundary condition appears at the GUT scale or above it. With the boundary conditions we have chosen, we accurately calculate the intermediate scale effects on the EDMs. We also discuss the b-r unification hypothesis for SUSY GUTS with an intermediate gauge symmetry breaking scale. Whenever in SUSY SO( 10) grand unification without an intermediate scale a tau neutrino mass is desired in the interesting eV range, it is found that b-r unification hypothesis has to be abandoned [7]. With the introduction of the intermediate scale, we examine how well that hypothesis works for the various models considered here. We know that the intermediate scale gauge symmetry breaking theories with SU( 2)~ x SU( 2) R x SU(4)c [S] or Sum x Sum x U(l)s__~ x SU(3)c [3] as the intermediate scale gauge group, give rise to large Iepton flavor violation which could be detected through processes like p -+ ey. The reason is quite simple. With intermediate gauge symmetry Sum x Sum x SU(4)c, the quarks and leptons are unified. Hence, the r-neutrino Yukawa coupling is the same as the top Yukawa coupling. Through the renormalization group equations (RGEs), the effect of the large r-neutrino Yukawa coupling is to make the third generation sleptons lighter than the first two generations, thus mitigating the GIM cancellation in one-loop leptonic flavor changing processes involving virtual sleptons. Although the quarks and leptons are not unified beneath the GUT scale when the intermediate scale gauge group is SU( 2)~ x SU( 2)~ x U(~)B__LXSU(~)C, thesameeffectisproducedfrom the assumption that the top quark Yukawa coupling is equal to the T-neutrino Yukawa coupling at the GUT scale. When we calculate the EDM of the electron the above stated principle applies, but we must also consider the phases at the gaugino-slepton-lepton vertices. Likewise, to generate the EDM for the neutron one needs the third generation down squark to be
Letters B 388 (1996) 605-610
lighter than those of the other two generations, which occurs due to the large top Yukawa coupling, and new phases at the gaugino-squark-quark vertices. In fact, whenever there is an intermediate scale, irrespective of the intermediate gauge group (e.g. SU( 2)~ x SU(2)~xSu(4)c orSU(2)LxSU(2)RxU( l)B_Lx SU(3)c) such phases are generated. The reason for this is that right-handed quarks or leptons are unified in a multiplet in a given generation. The superpotential for an intermediate gauge symmetry breaking model can be written (in the case of Sum x SU(2)R x SU(4)c): WY = &,F@
+ &,F@,F,
(1)
where F and E are the superfields containing the standard model fermion fields and transform as (2, 1, 4) and (1, 2, 4) respectively and we have suppressed the generation and gauge group indices. For simplicity we choose the MSSM Higgs Hi comes purely from bidoublet 41 and HZ from the bidoublet 42. However a linear combination of 41 and 42 can generate Ht and Hz, but that would not change the phase structure described below. Also in the small tan p case, which we will use, the most natural situation is that Hi comes naturally from &. This superpotential requires unification of lepton and down quark masses at the intermediate scale. We expect small non renormalizable terms to break this equality for the light quarks without affecting our main results. We shall work in a basis where hF, is diagonal in which WY can be expressed as the following: WY = F& ” i7Q2 +- FLJ*&dUtF@l.
(2)
The matrix ZJ is a general 3 x 3 unitary matrix with 3 angles and 6 phases. It can be written as follows: u=s’*vs,
(3)
where V is the CKM matrix and S and S’ are diagonal phase matrices. At the weak scale the superpotential for the Yukawa coupling can be expressed in the following manner: WMSSM= Q&,UcH2 + QV*&S2VtDCH~ + ECV;&,S2VfLHI,
(4)
N.G. Deshpande et al./Physics
where the ability to reduce the number of phases by redefinition of fields has been taken advantage of to the fullest extent possible,
s2 ZZ
e’+d 0 0 0 e+s 0 01 1 ( 0
(5)
is a diagonal phase matrix with two independent phases, and VI is the CKM matrix at the intermediate gauge symmetry breaking scale. Note that in the above equation the second term has the CKM matrix V at the low energy scale, while in the last term the leptonic mixing matrix stops running at the scale MI. It is not possible to do a superfield rotation on DC or L to remove the right handed angle since at MI the third diagonal element of the scalar mass matrices nzi and rni develop differently from the other two diagonal elements due to large top Yukawa coupling RGE effects. When the intermediate gauge symmetry group is Sum x Sum x U(~)B__L x SU(3)c, the additional CKM-like phases will be generated in exactly the same way as described above. The expressions from which we calculate EDMs are given below [4]. For the electron’s EDM we have l&l = elF2IIl$JVI,,,II
sin$I,
(6)
where Fz is given by F2=
ff 4~ COSMIC m~v",v,e,(V=;*>2(A,+pttanj?)
[Gz(m&,&) - G - G(m&,m&)+G(m&,m~,)l , x
(7)
and Vib are the matrix elements of the matrix VI and the functions G2 (a, b) are defined in Eq. (20) in Ref. [ 51. qbincludes effects of all possible phases. For the neutron l&l = 4/9eIF,‘II sin41,
(8)
where F2/is given by F,’ = ~vrdK~(v;“b)2(&
+ PtanP)
x lG2(mi,,m&) - G2(m&,miR)
- &(m%,m&) +
G(m~L,m~R)l.
(9)
To calculate the squark, slepton and gaugino masses at the low scale we numerically run the RGEs from
Letters B 388 (1996) 605-610
607
the GUT scale down to weak scale. We assume a universal boundary condition at the GUT scale MG i.e. all gaugino masses Mi( MG) = ml, dl tri-hXU SC&r couplings Ai( MG) = Ao, and ali soft scalar masses rnz(MG) = rng. We use the RGEs for a given intermediate gauge group from MG down to MI as given in Refs. [ 3,8]. From MI scale down to the weak scale, we use the MSSM RGEs (see, for example, Ref. [ 141) . As examples to illustrate our point, we choose to use the following four intermediate scale models: Model ( 1) : It is based on the gauge group SU( 2) R x SU(~>L x SU(4)c [81 with gauge couplings that are found to be unified at a scale MG near the string unification scale. The model breaks to the minimal supersymmetric standard model at a scale MI - 1012GeV and can have both large and small tanp scenarios. For high tan/? scenario we use MI = lOI2 GeV and MG M 1O’8.26GeV leading to aS(Mz) M 0.126 and for low tan p scenario we use MI = 1012 GeV and n/r,% 10*7.83GeV leading to au,( Mz) M 0.119. Model (2):ItisCaseVofRef. [9] whereSO(10) is broken down to Sum x SU(2)R x U( ~)B__L x SU( 3) c gauge symmetry at the scale MG. In this scenario, we use MI M 1012 GeV and MG M 1015.6 GeV leading to a,( Mz) M 0.129. Model (3) : This is the model presented in Ref. [ lo] with SO( 10) breaking down to SU(2)t x SU(2)R x U( ~)B__Lx SU(3)c. It is the onIy example we use for which D-parity is not broken at MG and hence leftright parity (gL = gR) is preserved in GI. The field content allows MI N 1 TeV with MG x 1016 GeV. We use MSSM below the scale MI for convenience, although in the original work [ lo] the two Higgs doublet model has been used. The value of cy, with the MSSM below MI is about 0.129. Model (4) : This is the model discussed in Ref. [ 111. Once again, SO( 10) is broken down to Sum X su(2)R X U( ~)B-_L X su(3)c at MG. In this model, MG is predicted to be exactly the same as in the conventional SUSY SO( 10) breaking with no intermediate scale and the scale MI can have any value between the TeV and the GUT scales. Since there is only one Higgs bidoublet, this model prefers large values of tanp with At = Ab at MI. Nevertheless, the introduction of nonrenormalizable operators can allow for small tan /3. In the Figs. 1 (a) -1 (d) , we plot
N.G. Deshpande et al./ Physics ‘Letters B 388 (1996) 605-610
608
(4
mg/GeV
m#GeV
w
(4 0.6t'
__----__
7
t 0
100
I 200
300
rj
,. 4
6
8
10
12
14
16
400
mdGeV Fig. 1. (a) : d, E Log to ( de/ (4.27. 1O-27 sin 4) ) for the low tan p version of model ( 1) is plotted as a function of the universal soft SUSY breaking mass mu. The solid lines correspond to p > 0, while the dashed lines correspond to p < 0. The upper two lines in the vicinity of WQ = 100 GeV are for rnr/a = 100 GeV, and the lower two lines are for ml/a = 150 GeV. At, = 3 54 for all the lines. (b): d, for model (2) is plotted as a function of the universal soft mass mu. The solid lines correspond to p > 0, while the dashed lines correspond to p < 0. The upper two lines in the vicinity of “10 = 150 GeV are for nr,a = 160 GeV, and the lower two lines are for rnllz = 200 GeV. At, = 3.54 for all the lines. (c): dr for Model (3) is plotted as a function of the universal soft SUSY breaking gaugino mass ml/a. The solid lines correspond to p > 0, and the dashed lines correspond to p < 0. The upper two lines around mo = 150 GeV are for BZ,,~ = 190 GeV, and the lower two lines in that region are for rnt,a = 220 GeV. Afc = 3.54 for all the lines. (d): dr for Model (4) is plotted as a function of logto Ml/GeV. The solid lines correspond to p > 0, and the dashed lines correspond to /L < 0. The upper two lines around MI = lo* GeV correspond to Ai0 = 3.54 and the lower two lines in the same region correspond to ArG = 1.38. ~YZQ = nrr,a = 180 GeV for all the lines.
(10)
where de is the EDM of electron, as a function of the scalar mass pfzofor different values of q/2. Since the EDM of neutron is also of the same order, we do not plot them. Also, experimentally the EDM of electron is more constrained. The experimental bounds are given as: d, < 0.8 . 1O-25 ecm [ 121 and d, <
4.3 . 1O-27 e cm [ 131. From the graphs it appears that one can use EDM as a signal for the intermediate scale in a grand unification scenario. The value of 1sin ~$1 could have arbitrary values from 0 to 1, but there is no reason to expect it to be suppressed. Now we discuss the viability of b --7 unification hypothesis in the models we have discussed. The value for rnrle from the existing data is w$“~ = 4.75 Z!Z0.05 calculated in Ref. [ 151, and we use mT = 1.777 GeV.
N.G. Deshpande et al. / Physics Letters B 388 (1996) 605-610
I
609
.
4
6
8
10
12
14
1
(b)
tan[beta]
Fig. 2. (a): The ml,po1e values are plotted as a function
of logI
MI/GeV
in model (4) with complete
unification. The solid line corresponds to (Ye= 0.117, sin’ BW = 0.2332 (within 2a of the experimental Mx scale, the dashed line corresponds to LYE= 0.122, sin2 0~ = 0.2321 (the experimental mid-value)
third generation
Yukawa coupling
mid-value) and c = l/127.9 at the and (Y= l/127.9 at the A42 scale.
AI, = 3.54 for both of the lines. (b): The mrle values are plotted as a function of tan p in model (4) with MI = lo’* GeV. The solid and dashed lines correspond to the same values of cu, as in Fig. 2(a). At, = 3.54 for both of the lines.
The predicted mb mass in these intermediate scale models mainly depend on 3 factors: the value of Ate, (Y$( Mz) and the location of the intermediate scale MI. Using larger values of A, of course lowers the mb mass, while using larger values of cu, increases it. For these models at the scale MG, we have used the maximum perturbative value for the top Yukawa coupling which is about 3.54. For model ( 1)) since leptons and down quarks are unified in the same multiplet at the intermediate scales we have hb = A, # At = hvr for the low tan p scenario. We find mpbDle = 4.78 GeV. For
the large tan/I version of that model, we have At = &, = A, = A,,, instead, and find mpbOle = 4.80 GeV. We find that model (1) is able to provide very reasonable b-quark mass predictions since czSis of moderate values and since down to the scale MI the relation Ab = A, exists intact. For models (2) and (3) with low values for tan& we have At = A, and & = A7 only at the GUT scale and find mb pole masses of 5.76 GeV and 6.20 GeV, respectively. However if we had used smaller values of LY,as used in the original Refs. [ 9,101 for those models or had we assumed large
610
N.G. Deshpande et al./Physics
values for tan p, these masses would be much closer to the desired range. As in Ref. [ 161, one could purposefully construct models with MI N lOI GeV and lower values for LY,so as to improve the b-quark mass prediction. For model (4) with AI = A&= A, = A,7 at MG, in Fig. 2(a) we choose to plot the rnpbOle mass as a function of MI since the intermediate gauge symmetry breaking scale in that model can lie anywhere between the weak scale and the GUT scale. Notice that the b-quark mass at first increases as the intermediate gauge symmetry breaking scale moves away from the GUT scale. But, it then reaches a peak value when the intermediate scale is about 10’ GeV, and then for MI less than that scale it decreases. The reason for this behavior can be found in the RGEs for Ab and A,. The RGE for Ab feels the influence of the large top Yukawa coupling while A, instead feels the influence of the r-neutrino coupling. Though the magnitude of the top and the r-neutrino couplings are same at the GUT scale, the r-neutrino coupling decreases faster than the top Yukawa coupling and reaches its fixed point sooner. If the intermediate breaking scale is decreased Ab( MI) would also decrease, however A,( MI) would not decrease as much. So, effectively the mass of mb decreases, since mb mass depends on the ratio of &, to A,. We further note that the interesting values for the intermediate gauge symmetry breaking scale MI N 1 TeV and MI N 1012GeV can both give good values for the b-quark mass. Effects of this low intermediate scale could be observed in the future colliders. In Fig. 2(b), we assume the possibility of model (4) allowing a range of values for tan ,&Iin order to plot the mpbOleas a function of tanp for the interesting case of MI = 1012 GeV. We see that larger values of tan p are preferred and give very reasonable values for the b-mass. In both Figs. 2(a) and 2(b), we show results for two different values of a3 as explained in the figure caption. In conclusion, we find that intermediate gauge symmetry breaking can be .a significant source of electric dipole moments for the electron and neutron. We have illustrated this effect for four different models. One of which has the gauge couplings unified at thestring scale and the others at the usual GUT scale
Letters B 388 (1996) 605-610
(N lOI6 GeV) . In all the models, the universal SUSY soft breaking boundary condition is assumed to be introduced at the GUT scale MG. Of course for the models where the gauge unification scale is of order 1016 GeV, if we had assumed the boundary condition at the reduced Planck scale or string scale the EDM predictions would have been further enhanced. We also examined how well the b-r unification hypothesis works for these models, and find that sometimes it works very well especially when the intermediate gauge symmetry unifies quarks and leptons or when tan /3 is large. We thank E. Ma for very useful discussions. This work was supported by Department of Energy grants DE-FG06-854ER 40224 and DE-FGO2-94ER 40837.
References [ 11 R. Shafer and Q. Shafi, Nature (London) 359 (1992) 199. [2] M. Dine, W. Fischler and M. Srednicki, Phys. Lett. B 104 (1981) 199. 131 N.G. Deshpande, B. Dutta and E. Keith, hep-ph/9512398 (to appear in Phys. Rev. D as a Rapid Comm.). t41 S. Dimopoulos and L.J. Hall, Phys. Lett. B 344 (1995) 185. I51 R. Barbieri, L.J. Hall and A. Strumia, Nucl. Phys. B 449 (1995) 437. [61 R. Barbieri, L.J. Hall and A. Strumia, Nucl. Phys. B 445 (1995) 219. r71 E Vissani and A.Y. Smirnov, Phys. L&t. B 341 (1994) 173; A. Brignole, H. Murayama and R. Rattazzi, Phys. Lett. B 335 (1994) 345. 181 N.G. Deshpande, B. Dutta and E. Keith, hep-ph/9604236 (to appear in Phys. Lett. B) 191 D.-G. Lee and R.N. Mohapatra, Phys. Rev. D 52 (1995) 4125. [lOI E. Ma, Phys. Rev. D 51 (1995) 236. Cl11 N.G. Deshpande, E. Keith and T.G. Rizzo, Phys. Rev. L&t. 70 (1993) 3189. [I21 Altarev et al., Phys. Lett. B 276 (1992) 242. [I31 Commins et al., LBL-35572. 1141 See, for example, V. Barger, M. Berger and P Ohmann, Phys. Rev. D 47 (1993) 1093; Phys. Rev. D 49 (1994) 4908. (1993) and 1151 M. Neubert, SLAC preprint SLAC-PUB-6263 references therein. [I61 B. Brahmachari and R.N. Mohapatra, Phys. Lett. B 357 (1995) 566.
21 November 1996
E!ld 33
ELSEVIER
PHYSICS
LETTERS B
Physics Letters B 388 (1996) 605-610
Electric dipole moments and b--7 unification in the presence of an intermediate scale in SUSY grand unification N.G. Deshpande a, B. Duttaa, E. Keith b a Institute of Theoretical Science, University of Oregon, Eugene, OR 97403, USA b Department of Physics, University of California, Riverside, CA 92521, USA
Received 8 June 1996; revised manuscript received 25 July 1996 Editor: M. Dine
Abstract We show that an intermediate gauge symmetry breaking scale can be a significant source of electric dipole moments for the electron and neutron in supersymmetric grand unified theories. New phases, similar to that of the CKM matrix, appear which do not arise from the supersymmetry (SUSY) breaking operators. To illustrate, we choose some grand unified SUSY models having an intermediate gauge symmetry breaking scale with some attractive features. We also show how well the b--7 unification hypothesis works in this class of models. PACS: 11.3O.Pb; 12.1O.Kt; 12.6O.J~; 14.60.Cd Keywords: Supersymmetry; Grand unification; Intermediate scale; Dipole moments
Supersymmetric grand unified theories (GUTS) with intermediate gauge symmetry breaking scales are attractive because they resolve a few long-standing problems and possess some desirable phenomenological features. For example, in models where the intermediate breaking scale MI N 10’“-10’2 GeV, one can naturally get a neutrino mass in the interesting range of N 3- 10 eV, which could serve as hot dark matter to explain the observed large scale structure formation of the universe [I]. The window N 10’“-10’2 GeV is also of the right size for a hypothetical PQ-symmetry to be broken so as to solve the strong CP problem without creating phenomenological or cosmological problems [ 21. Models which allow even lower intermediate gauge symmetry breaking scale, e.g. MI N 1 TeV are also interesting since they predict relatively light new gauge fields, as for example Sum charged gauge bosons WR. In all these intermediate 0370-2693/96/$12.00
scale models, lepton flavor violation is predicted [ 31 which may be close to the current experimental limit, and hence could provide a signal of such models. In this letter, we point out another special feature of these intermediate scale models: they can give rise to detectable amounts of electric dipole moments (EDMs) to the electron and neutron. This feature does not depend on the nature of the ultimate unifying group. We will always assume that supersymmetry is broken via soft breaking terms introduced at a super high scale. We shall assume that the soft breaking terms at the high scale at which they are introduced are flavor blind and CP invariant. It has already been shown [4-61 for SUSY SO( 10) models without an intermediate scale that there could be significant amounts of EDM for the electron and neutron. However for this to arise, the universal boundary condition for the soft SUSY breaking terms has to be imple-
Copyright 0 1996 Elsevier Science B.V. All rights reserved.
PII SO370-2693(96)01176-S
606
N.G. Deshpande et al./Physics
mented at a scale higher than the GUT scale MG such as the reduced Planck or string scales. Consequently, the EDMs for the electron and neutron are not expected to be produced in such a manner in SUSY GUTS with the attractive feature of gauge unification taking place at the string scale. The models with an intermediate gauge symmetry breaking scale however give rise to electron and neutron EDMs, irrespective of whether a universal soft SUSY breaking boundary condition appears at the GUT scale or above it. With the boundary conditions we have chosen, we accurately calculate the intermediate scale effects on the EDMs. We also discuss the b-r unification hypothesis for SUSY GUTS with an intermediate gauge symmetry breaking scale. Whenever in SUSY SO( 10) grand unification without an intermediate scale a tau neutrino mass is desired in the interesting eV range, it is found that b-r unification hypothesis has to be abandoned [7]. With the introduction of the intermediate scale, we examine how well that hypothesis works for the various models considered here. We know that the intermediate scale gauge symmetry breaking theories with SU( 2)~ x SU( 2) R x SU(4)c [S] or Sum x Sum x U(l)s__~ x SU(3)c [3] as the intermediate scale gauge group, give rise to large Iepton flavor violation which could be detected through processes like p -+ ey. The reason is quite simple. With intermediate gauge symmetry Sum x Sum x SU(4)c, the quarks and leptons are unified. Hence, the r-neutrino Yukawa coupling is the same as the top Yukawa coupling. Through the renormalization group equations (RGEs), the effect of the large r-neutrino Yukawa coupling is to make the third generation sleptons lighter than the first two generations, thus mitigating the GIM cancellation in one-loop leptonic flavor changing processes involving virtual sleptons. Although the quarks and leptons are not unified beneath the GUT scale when the intermediate scale gauge group is SU( 2)~ x SU( 2)~ x U(~)B__LXSU(~)C, thesameeffectisproducedfrom the assumption that the top quark Yukawa coupling is equal to the T-neutrino Yukawa coupling at the GUT scale. When we calculate the EDM of the electron the above stated principle applies, but we must also consider the phases at the gaugino-slepton-lepton vertices. Likewise, to generate the EDM for the neutron one needs the third generation down squark to be
Letters B 388 (1996) 605-610
lighter than those of the other two generations, which occurs due to the large top Yukawa coupling, and new phases at the gaugino-squark-quark vertices. In fact, whenever there is an intermediate scale, irrespective of the intermediate gauge group (e.g. SU( 2)~ x SU(2)~xSu(4)c orSU(2)LxSU(2)RxU( l)B_Lx SU(3)c) such phases are generated. The reason for this is that right-handed quarks or leptons are unified in a multiplet in a given generation. The superpotential for an intermediate gauge symmetry breaking model can be written (in the case of Sum x SU(2)R x SU(4)c): WY = &,F@
+ &,F@,F,
(1)
where F and E are the superfields containing the standard model fermion fields and transform as (2, 1, 4) and (1, 2, 4) respectively and we have suppressed the generation and gauge group indices. For simplicity we choose the MSSM Higgs Hi comes purely from bidoublet 41 and HZ from the bidoublet 42. However a linear combination of 41 and 42 can generate Ht and Hz, but that would not change the phase structure described below. Also in the small tan p case, which we will use, the most natural situation is that Hi comes naturally from &. This superpotential requires unification of lepton and down quark masses at the intermediate scale. We expect small non renormalizable terms to break this equality for the light quarks without affecting our main results. We shall work in a basis where hF, is diagonal in which WY can be expressed as the following: WY = F& ” i7Q2 +- FLJ*&dUtF@l.
(2)
The matrix ZJ is a general 3 x 3 unitary matrix with 3 angles and 6 phases. It can be written as follows: u=s’*vs,
(3)
where V is the CKM matrix and S and S’ are diagonal phase matrices. At the weak scale the superpotential for the Yukawa coupling can be expressed in the following manner: WMSSM= Q&,UcH2 + QV*&S2VtDCH~ + ECV;&,S2VfLHI,
(4)
N.G. Deshpande et al./Physics
where the ability to reduce the number of phases by redefinition of fields has been taken advantage of to the fullest extent possible,
s2 ZZ
e’+d 0 0 0 e+s 0 01 1 ( 0
(5)
is a diagonal phase matrix with two independent phases, and VI is the CKM matrix at the intermediate gauge symmetry breaking scale. Note that in the above equation the second term has the CKM matrix V at the low energy scale, while in the last term the leptonic mixing matrix stops running at the scale MI. It is not possible to do a superfield rotation on DC or L to remove the right handed angle since at MI the third diagonal element of the scalar mass matrices nzi and rni develop differently from the other two diagonal elements due to large top Yukawa coupling RGE effects. When the intermediate gauge symmetry group is Sum x Sum x U(~)B__L x SU(3)c, the additional CKM-like phases will be generated in exactly the same way as described above. The expressions from which we calculate EDMs are given below [4]. For the electron’s EDM we have l&l = elF2IIl$JVI,,,II
sin$I,
(6)
where Fz is given by F2=
ff 4~ COSMIC m~v",v,e,(V=;*>2(A,+pttanj?)
[Gz(m&,&) - G
(7)
and Vib are the matrix elements of the matrix VI and the functions G2 (a, b) are defined in Eq. (20) in Ref. [ 51. qbincludes effects of all possible phases. For the neutron l&l = 4/9eIF,‘II sin41,
(8)
where F2/is given by F,’ = ~vrdK~(v;“b)2(&
+ PtanP)
x lG2(mi,,m&) - G2(m&,miR)
- &(m%,m&) +
G(m~L,m~R)l.
(9)
To calculate the squark, slepton and gaugino masses at the low scale we numerically run the RGEs from
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the GUT scale down to weak scale. We assume a universal boundary condition at the GUT scale MG i.e. all gaugino masses Mi( MG) = ml, dl tri-hXU SC&r couplings Ai( MG) = Ao, and ali soft scalar masses rnz(MG) = rng. We use the RGEs for a given intermediate gauge group from MG down to MI as given in Refs. [ 3,8]. From MI scale down to the weak scale, we use the MSSM RGEs (see, for example, Ref. [ 141) . As examples to illustrate our point, we choose to use the following four intermediate scale models: Model ( 1) : It is based on the gauge group SU( 2) R x SU(~>L x SU(4)c [81 with gauge couplings that are found to be unified at a scale MG near the string unification scale. The model breaks to the minimal supersymmetric standard model at a scale MI - 1012GeV and can have both large and small tanp scenarios. For high tan/? scenario we use MI = lOI2 GeV and MG M 1O’8.26GeV leading to aS(Mz) M 0.126 and for low tan p scenario we use MI = 1012 GeV and n/r,% 10*7.83GeV leading to au,( Mz) M 0.119. Model (2):ItisCaseVofRef. [9] whereSO(10) is broken down to Sum x SU(2)R x U( ~)B__L x SU( 3) c gauge symmetry at the scale MG. In this scenario, we use MI M 1012 GeV and MG M 1015.6 GeV leading to a,( Mz) M 0.129. Model (3) : This is the model presented in Ref. [ lo] with SO( 10) breaking down to SU(2)t x SU(2)R x U( ~)B__Lx SU(3)c. It is the onIy example we use for which D-parity is not broken at MG and hence leftright parity (gL = gR) is preserved in GI. The field content allows MI N 1 TeV with MG x 1016 GeV. We use MSSM below the scale MI for convenience, although in the original work [ lo] the two Higgs doublet model has been used. The value of cy, with the MSSM below MI is about 0.129. Model (4) : This is the model discussed in Ref. [ 111. Once again, SO( 10) is broken down to Sum X su(2)R X U( ~)B-_L X su(3)c at MG. In this model, MG is predicted to be exactly the same as in the conventional SUSY SO( 10) breaking with no intermediate scale and the scale MI can have any value between the TeV and the GUT scales. Since there is only one Higgs bidoublet, this model prefers large values of tanp with At = Ab at MI. Nevertheless, the introduction of nonrenormalizable operators can allow for small tan /3. In the Figs. 1 (a) -1 (d) , we plot
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mdGeV Fig. 1. (a) : d, E Log to ( de/ (4.27. 1O-27 sin 4) ) for the low tan p version of model ( 1) is plotted as a function of the universal soft SUSY breaking mass mu. The solid lines correspond to p > 0, while the dashed lines correspond to p < 0. The upper two lines in the vicinity of WQ = 100 GeV are for rnr/a = 100 GeV, and the lower two lines are for ml/a = 150 GeV. At, = 3 54 for all the lines. (b): d, for model (2) is plotted as a function of the universal soft mass mu. The solid lines correspond to p > 0, while the dashed lines correspond to p < 0. The upper two lines in the vicinity of “10 = 150 GeV are for nr,a = 160 GeV, and the lower two lines are for rnllz = 200 GeV. At, = 3.54 for all the lines. (c): dr for Model (3) is plotted as a function of the universal soft SUSY breaking gaugino mass ml/a. The solid lines correspond to p > 0, and the dashed lines correspond to p < 0. The upper two lines around mo = 150 GeV are for BZ,,~ = 190 GeV, and the lower two lines in that region are for rnt,a = 220 GeV. Afc = 3.54 for all the lines. (d): dr for Model (4) is plotted as a function of logto Ml/GeV. The solid lines correspond to p > 0, and the dashed lines correspond to /L < 0. The upper two lines around MI = lo* GeV correspond to Ai0 = 3.54 and the lower two lines in the same region correspond to ArG = 1.38. ~YZQ = nrr,a = 180 GeV for all the lines.
(10)
where de is the EDM of electron, as a function of the scalar mass pfzofor different values of q/2. Since the EDM of neutron is also of the same order, we do not plot them. Also, experimentally the EDM of electron is more constrained. The experimental bounds are given as: d, < 0.8 . 1O-25 ecm [ 121 and d, <
4.3 . 1O-27 e cm [ 131. From the graphs it appears that one can use EDM as a signal for the intermediate scale in a grand unification scenario. The value of 1sin ~$1 could have arbitrary values from 0 to 1, but there is no reason to expect it to be suppressed. Now we discuss the viability of b --7 unification hypothesis in the models we have discussed. The value for rnrle from the existing data is w$“~ = 4.75 Z!Z0.05 calculated in Ref. [ 151, and we use mT = 1.777 GeV.
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mid-value) and c = l/127.9 at the and (Y= l/127.9 at the A42 scale.
AI, = 3.54 for both of the lines. (b): The mrle values are plotted as a function of tan p in model (4) with MI = lo’* GeV. The solid and dashed lines correspond to the same values of cu, as in Fig. 2(a). At, = 3.54 for both of the lines.
The predicted mb mass in these intermediate scale models mainly depend on 3 factors: the value of Ate, (Y$( Mz) and the location of the intermediate scale MI. Using larger values of A, of course lowers the mb mass, while using larger values of cu, increases it. For these models at the scale MG, we have used the maximum perturbative value for the top Yukawa coupling which is about 3.54. For model ( 1)) since leptons and down quarks are unified in the same multiplet at the intermediate scales we have hb = A, # At = hvr for the low tan p scenario. We find mpbDle = 4.78 GeV. For
the large tan/I version of that model, we have At = &, = A, = A,,, instead, and find mpbOle = 4.80 GeV. We find that model (1) is able to provide very reasonable b-quark mass predictions since czSis of moderate values and since down to the scale MI the relation Ab = A, exists intact. For models (2) and (3) with low values for tan& we have At = A, and & = A7 only at the GUT scale and find mb pole masses of 5.76 GeV and 6.20 GeV, respectively. However if we had used smaller values of LY,as used in the original Refs. [ 9,101 for those models or had we assumed large
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values for tan p, these masses would be much closer to the desired range. As in Ref. [ 161, one could purposefully construct models with MI N lOI GeV and lower values for LY,so as to improve the b-quark mass prediction. For model (4) with AI = A&= A, = A,7 at MG, in Fig. 2(a) we choose to plot the rnpbOle mass as a function of MI since the intermediate gauge symmetry breaking scale in that model can lie anywhere between the weak scale and the GUT scale. Notice that the b-quark mass at first increases as the intermediate gauge symmetry breaking scale moves away from the GUT scale. But, it then reaches a peak value when the intermediate scale is about 10’ GeV, and then for MI less than that scale it decreases. The reason for this behavior can be found in the RGEs for Ab and A,. The RGE for Ab feels the influence of the large top Yukawa coupling while A, instead feels the influence of the r-neutrino coupling. Though the magnitude of the top and the r-neutrino couplings are same at the GUT scale, the r-neutrino coupling decreases faster than the top Yukawa coupling and reaches its fixed point sooner. If the intermediate breaking scale is decreased Ab( MI) would also decrease, however A,( MI) would not decrease as much. So, effectively the mass of mb decreases, since mb mass depends on the ratio of &, to A,. We further note that the interesting values for the intermediate gauge symmetry breaking scale MI N 1 TeV and MI N 1012GeV can both give good values for the b-quark mass. Effects of this low intermediate scale could be observed in the future colliders. In Fig. 2(b), we assume the possibility of model (4) allowing a range of values for tan ,&Iin order to plot the mpbOleas a function of tanp for the interesting case of MI = 1012 GeV. We see that larger values of tan p are preferred and give very reasonable values for the b-mass. In both Figs. 2(a) and 2(b), we show results for two different values of a3 as explained in the figure caption. In conclusion, we find that intermediate gauge symmetry breaking can be .a significant source of electric dipole moments for the electron and neutron. We have illustrated this effect for four different models. One of which has the gauge couplings unified at thestring scale and the others at the usual GUT scale
Letters B 388 (1996) 605-610
(N lOI6 GeV) . In all the models, the universal SUSY soft breaking boundary condition is assumed to be introduced at the GUT scale MG. Of course for the models where the gauge unification scale is of order 1016 GeV, if we had assumed the boundary condition at the reduced Planck scale or string scale the EDM predictions would have been further enhanced. We also examined how well the b-r unification hypothesis works for these models, and find that sometimes it works very well especially when the intermediate gauge symmetry unifies quarks and leptons or when tan /3 is large. We thank E. Ma for very useful discussions. This work was supported by Department of Energy grants DE-FG06-854ER 40224 and DE-FGO2-94ER 40837.
References [ 11 R. Shafer and Q. Shafi, Nature (London) 359 (1992) 199. [2] M. Dine, W. Fischler and M. Srednicki, Phys. Lett. B 104 (1981) 199. 131 N.G. Deshpande, B. Dutta and E. Keith, hep-ph/9512398 (to appear in Phys. Rev. D as a Rapid Comm.). t41 S. Dimopoulos and L.J. Hall, Phys. Lett. B 344 (1995) 185. I51 R. Barbieri, L.J. Hall and A. Strumia, Nucl. Phys. B 449 (1995) 437. [61 R. Barbieri, L.J. Hall and A. Strumia, Nucl. Phys. B 445 (1995) 219. r71 E Vissani and A.Y. Smirnov, Phys. L&t. B 341 (1994) 173; A. Brignole, H. Murayama and R. Rattazzi, Phys. Lett. B 335 (1994) 345. 181 N.G. Deshpande, B. Dutta and E. Keith, hep-ph/9604236 (to appear in Phys. Lett. B) 191 D.-G. Lee and R.N. Mohapatra, Phys. Rev. D 52 (1995) 4125. [lOI E. Ma, Phys. Rev. D 51 (1995) 236. Cl11 N.G. Deshpande, E. Keith and T.G. Rizzo, Phys. Rev. L&t. 70 (1993) 3189. [I21 Altarev et al., Phys. Lett. B 276 (1992) 242. [I31 Commins et al., LBL-35572. 1141 See, for example, V. Barger, M. Berger and P Ohmann, Phys. Rev. D 47 (1993) 1093; Phys. Rev. D 49 (1994) 4908. (1993) and 1151 M. Neubert, SLAC preprint SLAC-PUB-6263 references therein. [I61 B. Brahmachari and R.N. Mohapatra, Phys. Lett. B 357 (1995) 566.